#10038491

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#57144	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4109 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

(3,9)(7,10)(11,15)(14,16)(17,18)(19,21)(20,26)(22,24)(23,25)(27,30)(28,31)(29,32)(33,34,39,36)(35,43,44,40)(37,41,47,45)(38,42,48,46);(1,2)(3,10)(4,8)(5,6)(7,9)(11,16)(12,13)(14,15)(17,19,23,22)(18,24,25,21)(20,28,29,27)(26,30,32,31)(33,35)(34,40)(36,43)(37,38)(39,44)(41,46)(42,45)(47,48);(2,6)(7,10)(8,12)(14,16)(18,25)(20,29)(21,24)(27,28)(33,34,39,36)(35,40,44,43)(37,45,47,41)(38,42,48,46);(1,3,5,9)(2,7,6,10)(4,11,13,15)(8,14,12,16)(17,20,23,29)(18,26,25,32)(19,27,22,28)(21,30,24,31)(33,36,39,34)(35,40,44,43)(37,45,47,41)(38,46,48,42);(1,2)(3,7)(4,12)(5,6)(8,13)(9,10)(11,16)(14,15)(17,21,23,24)(18,19,25,22)(20,30,29,31)(26,27,32,28)(33,37,39,47)(34,41,36,45)(35,38,44,48)(40,46,43,42);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,22,23,19)(18,21,25,24)(20,28,29,27)(26,30,32,31)(33,38)(34,42)(35,37)(36,46)(39,48)(40,45)(41,43)(44,47)

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=gh,d^2=gi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9945687);